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The EulerPoincaré equations and semidirect products with applications to continuum theories
 ADV. MATH
, 1998
"... We study Euler–Poincaré systems (i.e., the Lagrangian analogue of LiePoisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler–Poincaré equations for a parameter dependent Lagrangian by using a variational principle of Lagrange d’Alembert type. ..."
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Cited by 231 (89 self)
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We study Euler–Poincaré systems (i.e., the Lagrangian analogue of LiePoisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler–Poincaré equations for a parameter dependent Lagrangian by using a variational principle of Lagrange d’Alembert type. Then we derive an abstract KelvinNoether theorem for these equations. We also explore their relation with the theory of LiePoisson Hamiltonian systems defined on the dual of a semidirect product Lie algebra. The Legendre transformation in such cases is often not invertible; thus, it does not produce a corresponding Euler–Poincaré system on that Lie algebra. We avoid this potential difficulty by developing the theory of Euler–Poincaré systems entirely within the Lagrangian framework. We apply the general theory to a number of known examples, including the heavy top, ideal compressible fluids and MHD. We also use this framework to derive higher dimensional CamassaHolm equations, which have many potentially interesting analytical properties. These
The problem of integrable discretization: Hamiltonian approach
 Progress in Mathematics, Volume 219. Birkhäuser
"... this paper. Hence, the nature of this discretization as a member of the Toda hierarchy was not understood properly. A complete account was given in Suris (1995). The work Rutishauser (1954) contains not only the denition of the qd algorithm from the viewpoint of the numerical analist, but also the e ..."
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Cited by 71 (2 self)
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this paper. Hence, the nature of this discretization as a member of the Toda hierarchy was not understood properly. A complete account was given in Suris (1995). The work Rutishauser (1954) contains not only the denition of the qd algorithm from the viewpoint of the numerical analist, but also the equations of motion of the Toda lattice (under the name of a \continuous analogue of the qd algorithm")! The relation of the qd algorithm to integrable systems might have important implications for the numerical analysis, cf. Deift et al. (1991), Nagai and Satsuma (1995).
Discrete EulerPoincaré and LiePoisson equations
 Nonlinearity
, 1999
"... Abstract. In this paper, discrete analogues of EulerPoincaré and LiePoisson reduction theory are developed for systems on finite dimensional Lie groups G with Lagrangians L: TG → R that are Ginvariant. These discrete equations provide “reduced ” numerical algorithms which manifestly preserve the ..."
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Cited by 56 (7 self)
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Abstract. In this paper, discrete analogues of EulerPoincaré and LiePoisson reduction theory are developed for systems on finite dimensional Lie groups G with Lagrangians L: TG → R that are Ginvariant. These discrete equations provide “reduced ” numerical algorithms which manifestly preserve the symplectic structure. The manifold G × G is used as an approximation of TG,and a discrete Langragian L: G × G → R is constructed in such a way that the Ginvariance property is preserved. Reduction by G results in new “variational” principle for the reduced Lagrangian ℓ: G → R, and provides the discrete EulerPoincaré (DEP) equations. Reconstruction of these equations recovers the discrete EulerLagrange equations developed in [MPS 98, WM 97] which are naturally symplecticmomentum algorithms. Furthermore, the solution of the DEP algorithm immediately leads to a discrete LiePoisson (DLP) algorithm. It is shown that when G =SO(n), the DEP and DLP algorithms for a particular choice of the discrete Lagrangian L are equivalent to the Moser
A Lie group variational integrator for the attitude dynamics of a rigid body with applications to the 3D pendulum
 Proceedings of the IEEE Conference on Control Applications
, 2005
"... Abstract — A numerical integrator is derived for a class of models that describe the attitude dynamics of a rigid body in the presence of an attitude dependent potential. The numerical integrator is obtained from a discrete variational principle, and exhibits excellent geometric conservation propert ..."
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Cited by 49 (30 self)
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Abstract — A numerical integrator is derived for a class of models that describe the attitude dynamics of a rigid body in the presence of an attitude dependent potential. The numerical integrator is obtained from a discrete variational principle, and exhibits excellent geometric conservation properties. In particular, by performing computations at the level of the Lie algebra, and updating the solution using the matrix exponential, the attitude automatically evolves on the rotation group embedded in the space of matrices. The geometric conservation properties of the numerical integrator imply long time numerical stability. We apply this variational integrator to the uncontrolled 3D pendulum, that is a rigid asymmetric body supported at a frictionless pivot acting under the influence of uniform gravity. Interesting dynamics of the 3D pendulum are exposed. I.
Lie group variational integrators for the full body problem
, 2007
"... We develop the equations of motion for full body models that describe the dynamics of rigid bodies, acting under their mutual gravity. The equations are derived using a variational approach where variations are defined on the Lie group of rigid body configurations. Both continuous equations of motio ..."
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Cited by 45 (28 self)
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We develop the equations of motion for full body models that describe the dynamics of rigid bodies, acting under their mutual gravity. The equations are derived using a variational approach where variations are defined on the Lie group of rigid body configurations. Both continuous equations of motion and variational integrators are developed in Lagrangian and Hamiltonian forms, and the reduction from the inertial frame to a relative frame is also carried out. The Lie group variational integrators are shown to be symplectic, to preserve conserved quantities, and to guarantee exact evolution on the configuration space. One of these variational integrators is used to simulate the dynamics of two rigid dumbbell bodies.
Hamilton–Pontryagin integrators on Lie groups
, 2007
"... Abstract In this paper, structurepreserving timeintegrators for rigid bodytype mechanical ..."
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Cited by 37 (7 self)
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Abstract In this paper, structurepreserving timeintegrators for rigid bodytype mechanical
Finite element analysis of nonsmooth contact
, 1999
"... This work develops robust contact algorithms capable of dealing with complex contact situations involving several bodies with corners. Amongst the mathematical tools we bring to bear on the problem is nonsmooth analysis, following Clarke (F.H. Clarke. Optimization and nonsmooth analysis. John Wiley ..."
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Cited by 34 (11 self)
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This work develops robust contact algorithms capable of dealing with complex contact situations involving several bodies with corners. Amongst the mathematical tools we bring to bear on the problem is nonsmooth analysis, following Clarke (F.H. Clarke. Optimization and nonsmooth analysis. John Wiley and Sons, New York, 1983.). We specifically address contact geometries for which both the use of normals and gap functions have difficulties and therefore precludes the application of most contact algorithms proposed to date. Such situations arise in applications such as fragmentation, where angular fragments undergo complex collision sequences before they scatter. We demonstrate the robustness and versatility of the nonsmooth contact algorithms developed in this paper with the aid of selected two and threedimensional applications.
Nonholonomic integrators
, 2001
"... Abstract. We introduce a discretization of the Lagranged’Alembert principle for Lagrangian systems with nonholonomic constraints, which allows us to construct numerical integrators that approximate the continuous flow. We study the geometric invariance properties of the discrete flow which provide ..."
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Cited by 31 (0 self)
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Abstract. We introduce a discretization of the Lagranged’Alembert principle for Lagrangian systems with nonholonomic constraints, which allows us to construct numerical integrators that approximate the continuous flow. We study the geometric invariance properties of the discrete flow which provide an explanation for the good performance of the proposed method. This is tested on two examples: a nonholonomic particle with a quadratic potential and a mobile robot with fixed orientation.
A Variational Complex For Difference Equations
"... A variational complex for dierence equations is described that yields a characterization of dierence Euler Lagrange equations. The proof that the complex is locally exact is fully detailed. In particular, an analogue of the Poincare lemma for exact forms on a lattice is stated and proved, and homoto ..."
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Cited by 29 (3 self)
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A variational complex for dierence equations is described that yields a characterization of dierence Euler Lagrange equations. The proof that the complex is locally exact is fully detailed. In particular, an analogue of the Poincare lemma for exact forms on a lattice is stated and proved, and homotopy maps are given that allow one to calculate discrete Lagrangians for discrete EulerLagrange systems. Applications to the calculation of conservation laws and the continuum limit of our complex are outlined. 1 Contents 1
Stochastic variational integrators
 IMA Journal of Numerical Analysis Advance
, 2008
"... This paper presents a continuous and discrete Lagrangian theory for stochastic Hamiltonian systems on manifolds. The main result is to derive stochastic governing equations for such systems from a critical point of a stochastic action. Using this result the paper derives Langevintype equations for ..."
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Cited by 29 (1 self)
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This paper presents a continuous and discrete Lagrangian theory for stochastic Hamiltonian systems on manifolds. The main result is to derive stochastic governing equations for such systems from a critical point of a stochastic action. Using this result the paper derives Langevintype equations for constrained mechanical systems and implements a stochastic analog of Lagrangian reduction. These are easy consequences of the fact that the stochastic action is intrinsically defined. Stochastic variational integrators (SVIs) are developed using a discretized stochastic variational principle. The paper shows that the discrete flow of an SVI is a.s. symplectic and in the presence of symmetry a.s. momentummap preserving. A firstorder meansquare convergent SVI for mechanical systems on Lie groups is introduced. As an application of the theory, SVIs are exhibited for multiple, randomly forced and torqued rigidbodies interacting via a potential. 1